Related papers: Jet schemes, log discrepancies and Inversion of Ad…
We give a self-contained presentation of the basic results on jet schemes of singular varieties. Applications are given to invariants of singularities, such as minimal log discrepancies. We simplify our older approach to Inversion of…
This paper summarizes the results at the present moment about singularities with respect to the Mather-Jacobian log discrepancies over algebraically closed field of arbitrary characteristic. The basic point is the Inversion of Adjunction…
We formulate a comparison of minimal log discrepancies of a variety and its ambient space with appropriate boundaries in terms of motivic integration. It was obtained also by Ein and Musta\c{t}\v{a} independently.
We compute the minimal log discrepancies of determinantal varieties of square matrices, and more generally of pairs $\bigl(D^k,\sum \alpha_i D^{k_i}\bigr)$ consisting of a determinantal variety (of square matrices) and an $\mathbb R$-linear…
We prove a precise inversion of adjunction formula for the log pair associated to a non-degenerate hypersurface. As a corollary, the minimal log discrepancies of non-degenerate normal hypersurface singularities are bounded from above by…
We prove the precise inversion of adjunction formula for finite linear group quotients of complete intersection varieties defined by semi-invariant equations. As an application, we prove the semi-continuity of minimal log discrepancies for…
We compute the Mather minimal log discrepancy via jet schemes and arc spaces for toric varieties and very general hypersurfaces.
We prove the precise inversion of adjunction formula for quotient singularities and klt Cartier divisors. As an application, we prove the semi-continuity of minimal log discrepancies for klt hyperquotient singularities.
The theory of relative logarithmic jet spaces is developed for log schemes. With this theory the existence of bounds of intersection multiplicities of curves and divisors on certain log schemes is established. This result extends those of…
The goal of this paper is a classification theorem of the singularities according to a new invariant, Mather discrepancy. On the other hand, we show some evidences convincing us that Mather discrepancy is a considerable invariant: By…
We give a counterexample to the PIA (precise inversion of adjunction) conjecture for minimal log discrepancies. We also give a counterexample to the LSC conjecture for families.
We construct a theory of motivic integration for smooth rigid varieties. As an application new invariants of degenerations are obtained.
We introduce new notions of log jet spaces. Mildly singular spaces are ``smooth'' in log geometry, so their log jet spaces behave like the jet spaces of smooth varieties. Myriad examples contrast log jet spaces with the usual jet spaces of…
We prove that if X is a locally complete intersection variety, then X has all the jet schemes irreducible if and only if X has canonical singularities. After embedding X in a smooth variety Y, we use motivic integration to express the…
The minimal log discrepancy is an invariant of singularities that plays an important role in the birational classification of algebraic varieties. Shokurov conjectured that the minimal log discrepancy can always be bounded from above in…
Using the structure of the jet schemes of rational double point singularities, we construct "minimal embedded toric resolutions" of these singularities. We also establish, for these singularities, a correspondence between a natural class of…
We show the existence of prime divisors computing minimal log discrepancies in positive characteristic except for a special case. Moreover we prove the lower semicontinuity of minimal log discrepancies for smooth varieties in positive…
In this paper, we study the singularities of a pair (X,Y) in arbitrary characteristic via jet schemes. For a smooth variety X in characteristic 0, Ein, Lazarsfeld and Mustata showed that there is a correspondence between irreducible closed…
We present a general, functorial approach to Motivic Integration for separated schemes of finite type in lieu of recent work by Hans Schoutens on the subject. Presented is a change of variables formula and a hierarchy of stability…
We establish adjunction and inversion of adjunction for log canonical centers of arbitrary codimension in full generality.